What is the integral of (x^8)/((2-x^9)^2) ?

The integral of (x^8)/((2-x^9)^2) is 1/(9(2-x^9))+C

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integral of (x^8)/((2-x^9)^2)

\int \frac{x ^{8}}{( 2 - x ^{9} ) ^{2}} d x

\frac{1}{9 ( 2 - x ^{9} )} + C

Solution steps

\int \frac{x ^{8}}{( 2 - x ^{9} ) ^{2}} d x

Apply u-substitution

: \int - \frac{1}{9 u ^{2}} d u

= \int - \frac{1}{9 u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{9} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{9} (- \frac{1}{u})

Substitute back

u = 2 - x^{9}

= - \frac{1}{9} (- \frac{1}{2 - x ^{9}})

Simplify

- \frac{1}{9} (- \frac{1}{2 - x ^{9}}) : \frac{1}{9 ( 2 - x ^{9} )}

= \frac{1}{9 ( 2 - x ^{9} )}

Add a constant to the solution

= \frac{1}{9 ( 2 - x ^{9} )} + C

x (\frac{d y}{d x}) - y = x^{2} sin (x)

y d x + (x - 1) d y = 0, y (2) = 9

y = \frac{x ^{2} + 8 x + 3}{x}

\frac{d}{d x} (\frac{g ( x ) ( x )}{a x ^{2} + x})

\frac{7}{x ^{2} + 3}

What is the integral of (x^8)/((2-x^9)^2) ?
The integral of (x^8)/((2-x^9)^2) is 1/(9(2-x^9))+C